Parity Sheaves

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Parity Sheaves

Olivier Dudas and Peng Shan

Sheaves in Representation Theory Isle of Skye

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Introduction

Motivation. Use geometric and homological methods in modular representation theory, namely sheaves with coefficients in a field k

◮ char(k) =0: Kazhdan-Lusztig conjecture can be proved usingIC complexes andthe decomposition theorem

◮ char(k) =p: IC complexes exist but the decomposition theorem is no longer true

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Introduction

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Introduction

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Introduction

Observation. In the flag variety (as in many other situations) Hⁱ IC(Xw,k)=0 unlessi+ℓ(w)is even

when char(k) =0. But not always true in prime characteristic (torsion).

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Failure of the decomposition theorem in sl

₂

Springer resolution of N ={(a,b,c) ∈C³|a²−bc =0}

•

Stalks ofRπ_∗(k[2])

−2 −1 0 Oreg k 0 0 {0} k 0 k

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Failure of the decomposition theorem in sl

₂

•

−2 −1 0 Oreg k 0 0 {0} k 0 k

Stalks of IC(N,k) ifchar(k) 6=2

−2 −1 0 O_reg k 0 0 {0} k 0 0

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Failure of the decomposition theorem in sl

₂

•

−2 −1 0 Oreg k 0 0 {0} k 0 k

Stalks of IC(N,k) ifchar(k) =2

−2 −1 0 O_reg k 0 0 {0} k k 0

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Parity complexes

Setting. X complex algebraic variety with Whitney stratification X = ^a

λ∈Λ

X_λ

Dc(X,k) =derived category of constructible k-sheaves Definition

A even sheafF is any bounded complex inDc(X,k) such that Hⁱ(F) =Hⁱ(DF) =0 for odd i

Equivalently, the stalks and costalks are concentrated in even degrees.

F is parity if it is a direct sumF^′⊕ F^′′[1]withF^′ andF^′′ even.

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Parity complexes

Setting. X complex algebraicG-variety with Whitney stratification X = ^a

λ∈Λ

X_λ

Dc(X,k) =derived category of G-equivariant constructiblek-sheaves Definition

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Parity complexes

λ∈Λ

X_λ

Dc(X,k) =derived category of constructible k-sheaves Definition

Remark. can adapt the definition whenk is a local ring.

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IC complexes vs parity sheaves

Aim. Find a good substitute for IC complexes in prime characteristic

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IC complexes vs parity sheaves

Aim. Find a good substitute for IC complexes in prime characteristic Main features of ICs

◮ Simple perverse sheaves up to shift are IC(Xλ,L) which extend irreducible local systems L[dim(Xλ)]

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IC complexes vs parity sheaves

◮ DIC(X_λ,L)≃IC(X_λ,L^∨)

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IC complexes vs parity sheaves

◮ Existence of ICs for any simple local system (via Deligne’s construction)

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IC complexes vs parity sheaves

◮ Decomposition theorem in

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IC complexes vs parity sheaves

◮ Decomposition theorem in char. 0

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Indecomposable parity complexes

λ∈Λ

X_λ

satisfying a parity vanishing condition

Hⁱ(X_λ,L_|X_λ) =0 for odd i and any local systemL

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Indecomposable parity complexes

Setting. X complex algebraicG-variety with Whitney stratification X = ^a

λ∈Λ

X_λ

HⁱG(X_λ,L_|X_λ) =0 for odd i and any local systemL

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Indecomposable parity complexes

λ∈Λ

X_λ

Examples

◮ (Kac-Moody) Flag varieties stratified by the Schubert cells (including the affine Grassmannian)

◮ G-orbits in the nilpotent cone ofg

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Indecomposable parity complexes

λ∈Λ

X_λ

Examples

◮ (Kac-Moody) Flag varieties stratified by the Schubert cells (including the affine Grassmannian)

◮ G-orbits in the nilpotent cone ofg

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Indecomposable parity complexes (2)

Theorem (Juteau-Mautner-Williamson)

Given an indecomposable local systemL onX_λ there exists, up to isomorphism, at most one indecomposable parity complex E(Xλ,L) such that

◮ E(Xλ,L) is supported onX_λ

◮ E(Xλ,L)

^Xλ ≃ L[dimX_λ]

Moreover, any indecomposable parity complex is isomorphic to a shift E(Xλ,L)[m] for some indecomposable local system L.

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Indecomposable parity complexes (2)

◮ E(Xλ,L)

The parity complexes E(Xλ,L) are called parity sheaves

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Indecomposable parity complexes (2)

◮ E(Xλ,L)

The parity complexes E(Xλ,L) are called parity sheaves As a consequence DE(X ,L)≃ E(X ,L^∨)

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IC complexes vs parity sheaves

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IC complexes vs parity sheaves

Main features of parity sheaves

◮ Indecomposable parity complexes up to shift are E(Xλ,L) which extend irreducible local systems L[dim(Xλ)]

◮ DE(Xλ,L)≃ E(Xλ,L^∨)

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IC complexes vs parity sheaves

◮ DE(Xλ,L)≃ E(Xλ,L^∨)

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Even resolutions

A morphism π:Y =∐Yµ−→X =∐Xλ is astratified morphismif (i) eachπ⁻¹(Xλ) is a union of strata

(ii) each surjective restrictionY_µ−→X_λ is a fibration with smooth fibers

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Even resolutions

(ii) each surjective restrictionY_µ−→X_λ is a fibration with smooth fibers π is said to beeven if moreover

(iii) thefibersof Y_µ−→X_λ have cohomology concentrated in even degrees

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Even resolutions

Idea

Use pushforward along even proper map/resolution

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Even resolutions

Idea

Use pushforward along even proper map/resolution This gives existence of parity sheaves for

◮ Flag varieties stratified by Schubert cells

◮ GLn-orbits in the nilpotent cone of gln

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IC complexes vs parity sheaves

◮ DE(Xλ,L)≃ E(Xλ,L^∨)

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IC complexes vs parity sheaves

◮ DE(Xλ,L)≃ E(Xλ,L^∨)

◮ Existence of parity sheaves via even resolution (when such a resolution exists)

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IC complexes vs parity sheaves

◮ DE(Xλ,L)≃ E(Xλ,L^∨)

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Decomposition theorem

For proving the existence, one needs an ”analogue” of the decomposition theorem for parity sheaves:

Theorem-Observation (Juteau-Mautner-Williamson)

Ifπ :Y −→X is a proper even map andF a parity complex onY, then π_∗F is parity.

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Decomposition theorem

For proving the existence, one needs an ”analogue” of the decomposition theorem for parity sheaves:

Theorem-Observation (Juteau-Mautner-Williamson)

Ifπ :Y −→X is a proper even map andF a parity complex onY, then π_∗F is parity.

Consequences. ifY is smooth

π_∗kY[dimY] ≃ ^ME(Xλi,Lⁱ)[ci]

In particular, if char(k) =0, the usual decomposition theorem shows that the IC complexes occurring are parity sheaves.

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IC complexes vs parity sheaves

◮ DE(Xλ,L)≃ E(Xλ,L^∨)

◮ ”Decomposition theorem inany characteristic”